Real analysis is a branch of pure mathematics which forms the basis for many other subfields, such as calculus, differential equations, and probability. In turn, real analysis is based on fundamental concepts from number theory and topology. To study real analysis you need a solid background in calculus and a facility with logic and proofs.
The topics of real analysis include
Here are some questions to get you thinking about real analysis. First, what is a real number, anyway? If you think you know what "1" is, that's a start. The number 1 is a real number, as are the other natural, or counting numbers. Add to these zero and the negative numbers and you have the integers, all real numbers. There are more.
What about fractions? The set of rational numbers, such as 1/3 and -45/97 are also real numbers. But here we meet our first question. We like to think that 2/6 is the same as 1/3, even though they have different written expressions. How do we make this idea of "sameness" precise? And what about this expression: 0.3333333333...?
We haven't gotten to the irrational numbers yet, but their discovery by the ancient Greeks precipitated nothing short of a revolution in mathematics. What number x satisfies the equation x2 = 2? How do we know this number is real?
Thus, the real numbers comprise integers, rational numbers, and irrational numbers. You are probably familiar with the association of real numbers with points on a line, the real number line. Here's your final question: are there points on the line without real number names? That is, are there any holes in the line?
My real analysis course is organized around the Moore Method of learning. I provide definitions and statements of theorems, while students provide the proofs. The entire class (including myself) is available for consultation, congratulations, and constructive criticism. I ask my students not to look at textbooks while they are working on the course. But afterwards, some good ones to use for further study and reference are: